Development of a Framework to Characterise the Openness of Mathematical Tasks

Yeo, Joseph B. W.

doi:10.1007/s10763-015-9675-9

Cited by 46 publications

(51 citation statements)

References 12 publications

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“…It might be supposed that any varied diet of rich problemsolving tasks would automatically generate plentiful opportunities for students to gain practice in a multitude of important mathematical procedures, and that this would be a natural way for procedural fluency to be addressed in the curriculum. However, a rich, open-ended task may be approached in a variety of ways (Yeo, 2017), and, where a choice of approaches is possible, students may be drawn to those which utilise skills with which they are already familiar and comfortable. In this way, areas of weakness may remain unaddressed.…”

Section: Mathematical Etudes 21 Backgroundmentioning

confidence: 99%

Developing mathematical fluency: comparing exercises and rich tasks

Foster

2017

Educ Stud Math

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Achieving fluency in important mathematical procedures is fundamental to students' mathematical development. The usual way to develop procedural fluency is to practise repetitive exercises, but is this the only effective way? This paper reports three quasiexperimental studies carried out in a total of 11 secondary schools involving altogether 528 students aged 12-15. In each study, parallel classes were taught the same mathematical procedure before one class undertook traditional exercises while the other worked on a Bmathematical etude^(Foster International Journal of Mathematical Education in Science and Technology, 44(5), 765-774, 2013b), designed to be a richer task involving extensive opportunities for practice of the relevant procedure. Bayesian t tests on the gain scores between pre-and post-tests in each study provided evidence of no difference between the two conditions. A Bayesian meta-analysis of the three studies gave a combined Bayes factor of 5.83, constituting Bsubstantial^evidence (Jeffreys, 1961) in favour of the null hypothesis that etudes and exercises were equally effective, relative to the alternative hypothesis that they were not. These data support the conclusion that the mathematical etudes trialled are comparable to traditional exercises in their effects on procedural fluency. This could make etudes a viable alternative to exercises, since they offer the possibility of richer, more creative problem-solving activity, with comparable effectiveness in developing procedural fluency.

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Section: Mathematical Etudes 21 Backgroundmentioning

confidence: 99%

Developing mathematical fluency: comparing exercises and rich tasks

Foster

2017

Educ Stud Math

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“…If communication between two people is to be effective, they must have some shared understanding of the meaning of words that form the basis of their communication (Yeo, 2017). Teachers are regularly presented with advice on choosing the tasks that they use in their mathematics lessons, and this guidance often uses some of the adjectives discussed above, such as 'rich ' (e.g., DfE, 2014).…”

Section: Discussionmentioning

confidence: 99%

“…Researchers have developed language to categorise and describe desirable features of mathematics tasks, typically by defining words such as 'rich' (e.g., Ahmed, 1987;Griffin, 2009;McDonald & Watson, 2010;Stylianides & Stylianides, 2008;Swan, 2008;Yeo, 2007Yeo, , 2017. Whilst different definitions of a word like this usually have many commonalities, they are sometimes contrasting, or at least non-identical.…”

Section: Introductionmentioning

confidence: 99%

Teachers’ appraisals of adjectives relating to mathematics tasks

Foster

Inglis

2017

Educ Stud Math

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Curricular implementations are unlikely to deliver the anticipated benefits for mathematics learners if written guidance to teachers is interpreted and enacted differently from the ways that policymakers and curriculum designers intend. One way in which this could happen is in relation to the mathematics tasks that teachers deploy in the classroom. Teachers and curriculum designers have developed an extensive vocabulary for describing tasks, using adjectives such as 'rich', 'open', 'real-life', 'engaging' and so on. But do teachers have a shared understanding of what these adjectives mean when they are applied to mathematics tasks? In study 1, we investigated teachers' appraisals of adjectives used to describe mathematics tasks, finding that task appraisals vary on seven dimensions, which we termed engagement, demand, routineness, strangeness, inquiry, context and interactivity. In study 2, focusing on the five most prominent dimensions, we investigated whether teachers have a shared understanding of the meaning of adjectives when applied to mathematics tasks. We found that there was some agreement about inquiry and context, some disagreement about routineness and clear disagreement about engagement and demand. We conclude that at least some adjectives commonly used to describe tasks are interpreted very differently by different teachers. Implications for how tasks might be discussed meaningfully by teachers, teacher educators and curriculum designers are highlighted.

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“…Most tasks and problems are either open or closed. If the task is closed, then the answer is determinate, the goal is specific, there is only one possible method to solve it, and the task cannot be extended (Yeo, 2017). The openness in problems can appear in different ways (Pehkonen, 1997).…”

Section: Why Problem-solving and Open Problems?mentioning

confidence: 99%

How to promote learning in problem-solving?

Laine¹

2019

Implementation Research on Problem Solving in School Settings

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This article is based on my plenary talk at the joint conference of ProMath and the GDM working group on problem-solving in 2018. The aim of this article is to consider teaching and learning problem-solving from different perspectives taking into account the connection between 1) teacher’s actions and pupils’ solutions and 2) teacher’s actions and pupils’ affective reactions. Safe and supportive emotional atmosphere is base for students’ learning and attitudes towards mathematics. Teacher has a central role both in constructing emotional atmosphere and in offering cognitive support that pupils need in order to reach higher-level solutions. Teachers need to use activating guidance, i.e., ask good questions based on pupils’ solutions. Balancing between too much and too little guidance is not easy.

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Development of a Framework to Characterise the Openness of Mathematical Tasks

Cited by 46 publications

References 12 publications

Developing mathematical fluency: comparing exercises and rich tasks

Developing mathematical fluency: comparing exercises and rich tasks

Teachers’ appraisals of adjectives relating to mathematics tasks

How to promote learning in problem-solving?

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